Question

Two wires A and B are of the same materials. Their lengths are in the ratio $1:2$ and the diameters are in the ratio $2:1$, When stretched by force $F_A$ and $F_B$ respectively they get equal increase in their lengths. Then the ratio $\frac{F_A}{F_B}$ should be:

• A. $1:2$
• B. $1:1$
• C. $2:1$
• D. $8:1$

Answer 1

Answer: (D)

$8:1$

Using Hooke's law $Y=\frac{F/A}{\Delta L/L}$ where $A=\frac{\pi D^2}{4}$

$⟹$ $F=Y\frac{\pi D^2}{4}\frac{\Delta L}{L}$

As $Y$ and $\Delta L$ are constant, thus $F\propto \frac{D^2}{L}$

$⟹$ $\frac{F_A}{F_B}=\frac{D_A^2}{D_B^2}\times \frac{L_B}{L_A}$

Given : $D_A:D_B=2:1$ and $L_A:L_B=1:2$

$\therefore$ $\frac{F_A}{F_B}=\frac{4}{1}\times \frac{2}{1}=\frac{8}{1}$